Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be intimidating for new learners in their first years of high school or college.
Still, understanding how to deal with these equations is important because it is basic knowledge that will help them navigate higher mathematics and complex problems across different industries.
This article will share everything you should review to learn simplifying expressions. We’ll cover the proponents of simplifying expressions and then validate our comprehension via some sample problems.
How Does Simplifying Expressions Work?
Before you can learn how to simplify them, you must understand what expressions are in the first place.
In mathematics, expressions are descriptions that have at least two terms. These terms can contain numbers, variables, or both and can be connected through addition or subtraction.
To give an example, let’s take a look at the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two terms include both numbers (8 and 2) and variables (x and y).
Expressions containing coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is essential because it lays the groundwork for grasping how to solve them. Expressions can be written in convoluted ways, and without simplifying them, you will have a hard time attempting to solve them, with more opportunity for error.
Of course, each expression differ regarding how they are simplified based on what terms they incorporate, but there are general steps that can be applied to all rational expressions of real numbers, regardless of whether they are square roots, logarithms, or otherwise.
These steps are known as the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Resolve equations between the parentheses first by using addition or applying subtraction. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one inside.
Exponents. Where possible, use the exponent principles to simplify the terms that contain exponents.
Multiplication and Division. If the equation calls for it, use multiplication and division to simplify like terms that apply.
Addition and subtraction. Finally, add or subtract the remaining terms in the equation.
Rewrite. Make sure that there are no more like terms that need to be simplified, then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
Along with the PEMDAS principle, there are a few additional rules you must be aware of when simplifying algebraic expressions.
You can only simplify terms with common variables. When adding these terms, add the coefficient numbers and keep the variables as [[is|they are]-70. For example, the equation 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and retaining the x as it is.
Parentheses that include another expression on the outside of them need to utilize the distributive property. The distributive property prompts you to simplify terms on the outside of parentheses by distributing them to the terms inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the principle of multiplication. When two separate expressions within parentheses are multiplied, the distributive rule kicks in, and each separate term will will require multiplication by the other terms, making each set of equations, common factors of one another. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign directly outside of an expression in parentheses denotes that the negative expression must also need to be distributed, changing the signs of the terms inside the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign right outside the parentheses means that it will have distribution applied to the terms inside. However, this means that you are able to remove the parentheses and write the expression as is because the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The previous rules were easy enough to use as they only applied to principles that affect simple terms with numbers and variables. However, there are additional rules that you must apply when dealing with expressions with exponents.
Here, we will discuss the laws of exponents. 8 principles influence how we process exponents, that includes the following:
Zero Exponent Rule. This principle states that any term with the exponent of 0 equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 doesn't alter the value. Or a1 = a.
Product Rule. When two terms with equivalent variables are multiplied by each other, their product will add their two exponents. This is expressed in the formula am × an = am+n
Quotient Rule. When two terms with the same variables are divided by each other, their quotient applies subtraction to their applicable exponents. This is written as the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent is equivalent to the inverse of that term over 1. This is written as the formula a-m = 1/am; (a/b)-m = (b/a)m.
Power of a Power Rule. If an exponent is applied to a term that already has an exponent, the term will end up having a product of the two exponents that were applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that possess different variables should be applied to the respective variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will assume the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the rule that says that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions on the inside. Let’s witness the distributive property applied below.
Let’s simplify the equation 2(3x + 5).
The distributive property states that a(b + c) = ab + ac. Thus, the equation becomes:
2(3x + 5) = 2(3x) + 2(5)
The result is 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have multiple rules that you need to follow.
When an expression has fractions, here is what to keep in mind.
Distributive property. The distributive property a(b+c) = ab + ac, when applied to fractions, will multiply fractions one at a time by their denominators and numerators.
Laws of exponents. This tells us that fractions will more likely be the power of the quotient rule, which will subtract the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest state should be expressed in the expression. Use the PEMDAS property and be sure that no two terms have matching variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, logarithms, linear equations, or quadratic equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the properties that must be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to the expressions inside the parentheses, while PEMDAS will govern the order of simplification.
As a result of the distributive property, the term on the outside of the parentheses will be multiplied by the individual terms inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, be sure to add all the terms with matching variables, and all term should be in its lowest form.
28x + 28 - 3y
Rearrange the equation as follows:
28x - 3y + 28
Example 2
Simplify the expression 1/3x + y/4(5x + 2)
The PEMDAS rule expresses that the you should begin with expressions inside parentheses, and in this example, that expression also requires the distributive property. Here, the term y/4 should be distributed within the two terms within the parentheses, as follows.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for now and simplify the terms with factors attached to them. Remember we know from PEMDAS that fractions require multiplication of their denominators and numerators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple since any number divided by 1 is that same number or x/1 = x. Thus,
y(5x)/4
5xy/4
The expression y/4(2) then becomes:
y/4 * 2/1
2y/4
Thus, the overall expression is:
1/3x + 5xy/4 + 2y/4
Its final simplified version is:
1/3x + 5/4xy + 1/2y
Example 3
Simplify the expression: (4x2 + 3y)(6x + 1)
In exponential expressions, multiplication of algebraic expressions will be used to distribute all terms to each other, which gives us the equation:
4x2(6x + 1) + 3y(6x + 1)
4x2(6x) + 4x2(1) + 3y(6x) + 3y(1)
For the first expression, the power of a power rule is applied, meaning that we’ll have to add the exponents of two exponential expressions with the same variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Since there are no remaining like terms to apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you are required to obey PEMDAS, the exponential rule, and the distributive property rules in addition to the concept of multiplication of algebraic expressions. Finally, make sure that every term on your expression is in its most simplified form.
What is the difference between solving an equation and simplifying an expression?
Solving equations and simplifying expressions are vastly different, although, they can be incorporated into the same process the same process due to the fact that you first need to simplify expressions before solving them.
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